I stumbled over the following problem while trying to prove LDP's for branching processes. However, I state the problem more generally:
assume $(X_t)_t$ be a stochastic process, with $X_t\rightarrow 0$ as $t\rightarrow\infty$ a.s., further we know that $$ e^{at}X_t\rightarrow W>0 $$ almost surely, as $t$ goes to infinity. Now I am interested in the probabilities $$ \mathbb{P}(X_t \geq c). $$ The problem that concerns me is that I'd like to write $$ \mathbb{P}(X_t \geq c) \approx \mathbb{P}(e^{-at} W \geq c) $$ for $t$ sufficiently large.
I am pretty sure it is not that simple, since somehow I'd need more precise estimates on the convergence speed of $e^{at}X_t$. When trying to prove it, I used the following inequality for arbitrary $\varepsilon>0$ $$ \mathbb{P}(X_t \geq c) \leq \mathbb{P}(X_t \geq c, |X_t- e^{-at}W|< \varepsilon) + \mathbb{P}(|X_t- e^{-at}W|\geq \varepsilon) $$ which yields $$ \mathbb{P}(X_t \geq c) \leq \mathbb{P}(W +\varepsilon\geq ce^{at}) + \mathbb{P}(|e^{at}X_t- W|\geq \varepsilon e^{at}). $$ But this means that I need to know that the second term is sufficiently small and vanishes when plugging the probabilities into $\frac{1}{e^{at}}\log (\cdot)$, this is where the large deviation goal comes into play again.
I feel like having the wrong ansatz here or just thinking too complicated. I am happy about any kind of advice or reference to literature. Thanks a lot.